\(\int (c x)^m (a x^2+b x^3)^p \, dx\) [427]

Optimal result

Integrand size = 19, antiderivative size = 56 \[ \int (c x)^m \left (a x^2+b x^3\right )^p \, dx=\frac {c (c x)^{-1+m} \left (a x^2+b x^3\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,2+m+3 p,2+m+2 p,-\frac {b x}{a}\right )}{a (1+m+2 p)} \] Output:

c*(c*x)^(-1+m)*(b*x^3+a*x^2)^(p+1)*hypergeom([1, 2+m+3*p],[2+m+2*p],-b*x/a 
)/a/(1+m+2*p)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09 \[ \int (c x)^m \left (a x^2+b x^3\right )^p \, dx=\frac {x (c x)^m \left (x^2 (a+b x)\right )^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,1+m+2 p,2+m+2 p,-\frac {b x}{a}\right )}{1+m+2 p} \] Input:

Integrate[(c*x)^m*(a*x^2 + b*x^3)^p,x]
 

Output:

(x*(c*x)^m*(x^2*(a + b*x))^p*Hypergeometric2F1[-p, 1 + m + 2*p, 2 + m + 2* 
p, -((b*x)/a)])/((1 + m + 2*p)*(1 + (b*x)/a)^p)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1938, 76, 74}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c x)^m \left (a x^2+b x^3\right )^p \, dx\)

\(\Big \downarrow \) 1938

\(\displaystyle (c x)^m x^{-m-2 p} (a+b x)^{-p} \left (a x^2+b x^3\right )^p \int x^{m+2 p} (a+b x)^pdx\)

\(\Big \downarrow \) 76

\(\displaystyle (c x)^m x^{-m-2 p} \left (\frac {b x}{a}+1\right )^{-p} \left (a x^2+b x^3\right )^p \int x^{m+2 p} \left (\frac {b x}{a}+1\right )^pdx\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {x (c x)^m \left (\frac {b x}{a}+1\right )^{-p} \left (a x^2+b x^3\right )^p \operatorname {Hypergeometric2F1}\left (-p,m+2 p+1,m+2 p+2,-\frac {b x}{a}\right )}{m+2 p+1}\)

Input:

Int[(c*x)^m*(a*x^2 + b*x^3)^p,x]
 

Output:

(x*(c*x)^m*(a*x^2 + b*x^3)^p*Hypergeometric2F1[-p, 1 + m + 2*p, 2 + m + 2* 
p, -((b*x)/a)])/((1 + m + 2*p)*(1 + (b*x)/a)^p)
 

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 76

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart 
[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(b*x)^m*(1 + d* 
(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integer 
Q[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2 
^(-1)] && EqQ[c^2 - d^2, 0])) ||  !RationalQ[n])
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [F]

\[\int \left (c x \right )^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p}d x\]

Input:

int((c*x)^m*(b*x^3+a*x^2)^p,x)
 

Output:

int((c*x)^m*(b*x^3+a*x^2)^p,x)
 

Fricas [F]

\[ \int (c x)^m \left (a x^2+b x^3\right )^p \, dx=\int { {\left (b x^{3} + a x^{2}\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(b*x^3+a*x^2)^p,x, algorithm="fricas")
 

Output:

integral((b*x^3 + a*x^2)^p*(c*x)^m, x)
 

Sympy [F]

\[ \int (c x)^m \left (a x^2+b x^3\right )^p \, dx=\int \left (c x\right )^{m} \left (x^{2} \left (a + b x\right )\right )^{p}\, dx \] Input:

integrate((c*x)**m*(b*x**3+a*x**2)**p,x)
 

Output:

Integral((c*x)**m*(x**2*(a + b*x))**p, x)
 

Maxima [F]

\[ \int (c x)^m \left (a x^2+b x^3\right )^p \, dx=\int { {\left (b x^{3} + a x^{2}\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(b*x^3+a*x^2)^p,x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a*x^2)^p*(c*x)^m, x)
 

Giac [F]

\[ \int (c x)^m \left (a x^2+b x^3\right )^p \, dx=\int { {\left (b x^{3} + a x^{2}\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(b*x^3+a*x^2)^p,x, algorithm="giac")
 

Output:

integrate((b*x^3 + a*x^2)^p*(c*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c x)^m \left (a x^2+b x^3\right )^p \, dx=\int {\left (c\,x\right )}^m\,{\left (b\,x^3+a\,x^2\right )}^p \,d x \] Input:

int((c*x)^m*(a*x^2 + b*x^3)^p,x)
 

Output:

int((c*x)^m*(a*x^2 + b*x^3)^p, x)
 

Reduce [F]

\[ \int (c x)^m \left (a x^2+b x^3\right )^p \, dx=\frac {c^{m} \left (x^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p} a p +x^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p} b m x +3 x^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p} b p x -\left (\int \frac {x^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p}}{b \,m^{2} x^{2}+6 b m p \,x^{2}+9 b \,p^{2} x^{2}+a \,m^{2} x +6 a m p x +9 a \,p^{2} x +b m \,x^{2}+3 b p \,x^{2}+a m x +3 a p x}d x \right ) a^{2} m^{3} p -8 \left (\int \frac {x^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p}}{b \,m^{2} x^{2}+6 b m p \,x^{2}+9 b \,p^{2} x^{2}+a \,m^{2} x +6 a m p x +9 a \,p^{2} x +b m \,x^{2}+3 b p \,x^{2}+a m x +3 a p x}d x \right ) a^{2} m^{2} p^{2}-\left (\int \frac {x^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p}}{b \,m^{2} x^{2}+6 b m p \,x^{2}+9 b \,p^{2} x^{2}+a \,m^{2} x +6 a m p x +9 a \,p^{2} x +b m \,x^{2}+3 b p \,x^{2}+a m x +3 a p x}d x \right ) a^{2} m^{2} p -21 \left (\int \frac {x^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p}}{b \,m^{2} x^{2}+6 b m p \,x^{2}+9 b \,p^{2} x^{2}+a \,m^{2} x +6 a m p x +9 a \,p^{2} x +b m \,x^{2}+3 b p \,x^{2}+a m x +3 a p x}d x \right ) a^{2} m \,p^{3}-5 \left (\int \frac {x^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p}}{b \,m^{2} x^{2}+6 b m p \,x^{2}+9 b \,p^{2} x^{2}+a \,m^{2} x +6 a m p x +9 a \,p^{2} x +b m \,x^{2}+3 b p \,x^{2}+a m x +3 a p x}d x \right ) a^{2} m \,p^{2}-18 \left (\int \frac {x^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p}}{b \,m^{2} x^{2}+6 b m p \,x^{2}+9 b \,p^{2} x^{2}+a \,m^{2} x +6 a m p x +9 a \,p^{2} x +b m \,x^{2}+3 b p \,x^{2}+a m x +3 a p x}d x \right ) a^{2} p^{4}-6 \left (\int \frac {x^{m} \left (b \,x^{3}+a \,x^{2}\right )^{p}}{b \,m^{2} x^{2}+6 b m p \,x^{2}+9 b \,p^{2} x^{2}+a \,m^{2} x +6 a m p x +9 a \,p^{2} x +b m \,x^{2}+3 b p \,x^{2}+a m x +3 a p x}d x \right ) a^{2} p^{3}\right )}{b \left (m^{2}+6 m p +9 p^{2}+m +3 p \right )} \] Input:

int((c*x)^m*(b*x^3+a*x^2)^p,x)
 

Output:

(c**m*(x**m*(a*x**2 + b*x**3)**p*a*p + x**m*(a*x**2 + b*x**3)**p*b*m*x + 3 
*x**m*(a*x**2 + b*x**3)**p*b*p*x - int((x**m*(a*x**2 + b*x**3)**p)/(a*m**2 
*x + 6*a*m*p*x + a*m*x + 9*a*p**2*x + 3*a*p*x + b*m**2*x**2 + 6*b*m*p*x**2 
 + b*m*x**2 + 9*b*p**2*x**2 + 3*b*p*x**2),x)*a**2*m**3*p - 8*int((x**m*(a* 
x**2 + b*x**3)**p)/(a*m**2*x + 6*a*m*p*x + a*m*x + 9*a*p**2*x + 3*a*p*x + 
b*m**2*x**2 + 6*b*m*p*x**2 + b*m*x**2 + 9*b*p**2*x**2 + 3*b*p*x**2),x)*a** 
2*m**2*p**2 - int((x**m*(a*x**2 + b*x**3)**p)/(a*m**2*x + 6*a*m*p*x + a*m* 
x + 9*a*p**2*x + 3*a*p*x + b*m**2*x**2 + 6*b*m*p*x**2 + b*m*x**2 + 9*b*p** 
2*x**2 + 3*b*p*x**2),x)*a**2*m**2*p - 21*int((x**m*(a*x**2 + b*x**3)**p)/( 
a*m**2*x + 6*a*m*p*x + a*m*x + 9*a*p**2*x + 3*a*p*x + b*m**2*x**2 + 6*b*m* 
p*x**2 + b*m*x**2 + 9*b*p**2*x**2 + 3*b*p*x**2),x)*a**2*m*p**3 - 5*int((x* 
*m*(a*x**2 + b*x**3)**p)/(a*m**2*x + 6*a*m*p*x + a*m*x + 9*a*p**2*x + 3*a* 
p*x + b*m**2*x**2 + 6*b*m*p*x**2 + b*m*x**2 + 9*b*p**2*x**2 + 3*b*p*x**2), 
x)*a**2*m*p**2 - 18*int((x**m*(a*x**2 + b*x**3)**p)/(a*m**2*x + 6*a*m*p*x 
+ a*m*x + 9*a*p**2*x + 3*a*p*x + b*m**2*x**2 + 6*b*m*p*x**2 + b*m*x**2 + 9 
*b*p**2*x**2 + 3*b*p*x**2),x)*a**2*p**4 - 6*int((x**m*(a*x**2 + b*x**3)**p 
)/(a*m**2*x + 6*a*m*p*x + a*m*x + 9*a*p**2*x + 3*a*p*x + b*m**2*x**2 + 6*b 
*m*p*x**2 + b*m*x**2 + 9*b*p**2*x**2 + 3*b*p*x**2),x)*a**2*p**3))/(b*(m**2 
 + 6*m*p + m + 9*p**2 + 3*p))